Answer
Hyperboloid of two sheets
Work Step by Step
Here, we have: $-4x^2+y^2 -4z^2=4$
Comparing the above form with the equation $\displaystyle \frac{z^{2}}{c^{2}}=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$
we find that we have a hyperboloid of two sheets.
In this case, we see that when $z=k$, the traces are hyperbolas.
When $y=k$, then the traces are circles. That is, $k^2-4=4x^2+4z^2$
When $x=k$ (axis is the x-axis), the traces are hyperbolas.
Because the traces are hyperbolas and circles and have a gap at the center, we know that we have a Hyperboloid of two sheets.
